Abstract
In this paper it is shown that the relativistic analogue of the circulation in classical hydrodynamics is the integral
where i is the specific enthalpy of the fluid, U ∝ are the covariant components of the four velocity of the fluid and the integral is taken around a closed curve traveling with the fluid. This integral is similar to an integral given by Lichnerowicz, but differs in the factor multiplying U ∝ in the integrand.
The necessary and sufficient conditions for C to vanish are shown to be that the flow be irrotational and isentropic. The necessary and sufficient condition for C to be a constant of the motion is that the pressure be a function of the density alone at every point in space-time occupied by the fluid. This condition is shown to be violated when a shock is present. Results for similar integrals taken around vortex lines are also obtained. The relations between these results and Bernoulli's theorems are discussed.
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References
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Communicated by G. C. McVittie
This work was supported in part by the National Science Foundation.
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Taub, A.H. On circulation in relativistic hydrodynamics. Arch. Rational Mech. Anal. 3, 312–324 (1959). https://doi.org/10.1007/BF00284183
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DOI: https://doi.org/10.1007/BF00284183